The term ``biquad'' is short for ``bi-quadratic'', and is a common name for a two-pole, two-zero digital filter. The transfer function of the biquad can be defined as
As derived in §10.1.3, for real second-order polynomials having complex roots, it is often convenient to express the polynomial coefficients in terms of the radius and angle of the positive-frequency pole. For example, denoting the denominator polynomial by , we have
As discussed in §10.6.4, a common setting for the zeros when making a resonator is to place one at (dc) and the other at (half the sampling rate), i.e., and in Eq. (10.8) above . This zero placement normalizes the peak gain of the resonator if it is swept using the parameter.
Using the shift theorem for z transforms, the difference equation for the biquad can be written by inspection of the transfer function as
where denotes the input signal sample at time , and is the output signal. This is the form that is typically implemented in software. It is essentially the direct-form I implementation. (To obtain the official direct-form I structure, the overall gain must be not be pulled out separately, resulting in feedforward coefficients instead. See Chapter 9 for more about filter implementation forms.)